Ruby, Sapphire & Emerald Emerald-specific Stuff: Groudon and Kyogre | Good News: You can catch both Groudon and Kyogre in Emerald.
Bad News: It's ever so slightly harder than it was in Ruby/Sapphire (don't worry, it's only slightly).

If you want your own legendaries which can be used as tools for having people laugh at you for being a n00b (or just to complete your Pokedex, which is highly recommended. Honestly), then you'll need to have defeated the Pokemon League. Once that's done, return to the Weather Institution on Route 119 (that's the one to the West of Fortree City). Here, the man on the Second Floor will inform you that he's started monitoring the weather around Hoenn.

Whenever he mentions heavy rainfall, he's referring to the current location of Kyogre, as the following images show:

Possible Routes

Possible Location #1

Possible Location #2

Whenever he mentions a drought, he's referring to the current location of Groudon, as the following images show:

Possible Routes

Possible Location #1

Possible Location #2

Now, as you may have noticed, Kyogre will appear on either Route 105, 125, 127 or 129. Groudon's many interior homes are found on Routes 114, 115, 116 and 118. The above images can show you the locations of where to find them, since just to be awkward, there are two possible locations of the cave (yes, you're looking for a cave - be it a normal cave or an underwater diving spot in the case of Kyogre) on each Route. Groudon is found in Terra Cave, Kyogre is found in Marine Cave (in case you were wondering). Oh, and just to make life even more fun, unless you get there quickly enough, the cave will disappear. So as soon as Mr Weather Institute tells you where your target is, you're in a race against time. Don't worry though - you can always go back and get another location. Woo.

Once you're in the cave, it's so easy it should be classed as "easy".. which is what I just said. Go through the first door you come to into a room full of fog. Save now, because otherwise you'll only want to kill yourself if you fail to catch your elusive beast. Then approach the Pokemon (both are Level 70, oo-er), and attempt to catch. Simple as.